1
Joint Wireless Information and Energy Transfer
in a Two-User MIMO Interference Channel
arXiv:1303.1693v2 [cs.IT] 30 Apr 2013
Jaehyun Park, Member, IEEE, Bruno Clerckx, Member, IEEE,
Abstract
This paper investigates joint wireless information and energy transfer in a two-user MIMO interference channel, in which each receiver either decodes the incoming information data (information decoding,
ID) or harvests the RF energy (energy harvesting, EH) to operate with a potentially perpetual energy
supply. In the two-user interference channel, we have four different scenarios according to the receiver
mode – (ID1 , ID2 ), (EH1 , EH2 ), (EH1 , ID2 ), and (ID1 , EH2 ). While the maximum information
bit rate is unknown and finding the optimal transmission strategy is still open for (ID1 , ID2 ), we have
derived the optimal transmission strategy achieving the maximum harvested energy for (EH1 , EH2 ). For
(EH1 , ID2 ), and (ID1 , EH2 ), we find a necessary condition of the optimal transmission strategy and,
accordingly, identify the achievable rate-energy (R-E) tradeoff region for two transmission strategies
that satisfy the necessary condition - maximum energy beamforming (MEB) and minimum leakage
beamforming (MLB). Furthermore, a new transmission strategy satisfying the necessary condition signal-to-leakage-and-energy ratio (SLER) maximization beamforming - is proposed and shown to exhibit
a better R-E region than the MEB and the MLB strategies. Finally, we propose a mode scheduling method
to switch between (EH1 , ID2 ) and (ID1 , EH2 ) based on the SLER.
Index Terms
Joint wireless information and energy transfer, MIMO interference channel, Rank-one beamforming
J. Park is with the Broadcasting and Telecommunications Convergence Research Laboratory, Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea (e-mail: jhpark@ee.kaist.ac.kr)
B. Clerckx is with the Department of Electrical and Electronic Engineering, Imperial College London, South Kensington
Campus London SW7 2AZ, United Kingdom (e-mail:b.clerckx@imperial.ac.uk)
May 1, 2013
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2
I. I NTRODUCTION
Over the last decade, there has been a lot of interest to transfer energy wirelessly and recently, radiofrequency (RF) radiation has become a viable source for energy harvesting. It is nowadays possible to
transfer the energy wirelessly with a reasonable efficiency over small distances and, furthermore, the
wireless sensor network (WSN) in which the sensors are capable of harvesting RF energy to power their
own transmissions has been introduced in industry ( [1]–[4] and references therein).
The energy harvesting function can be exploited in either transmit side [5]–[9] or receive side [10]–
[13]. For the energy harvesting transmitter, energy harvesting scheduling and transmit power allocation
have been considered and, for the energy harvesting receiver, the management of information decoding
and energy harvesting has been developed. Furthermore, because RF signals carry information as well as
energy, “joint wireless information and energy transfer” in conjunction with the energy harvesting receiver
has been investigated [10]–[13]. That is, previous works have studied the fundamental performance limits
and the optimal transmission strategies of the joint wireless information and energy transfer in the cellular
downlink system with a single base station (BS) and multiple mobile stations (MSs) [12] and in the
cooperative relay system [13] and in the broadcasting system [10], [11] with a single energy receiver and
a single information receiver when they are separately located or co-located.
There have been very few studies of joint wireless information and energy transfer on the interference
channel (IFC) models [14]–[16]. In [14], [15], the authors have considered a two-user single-input singleoutput (SISO) IFC and derived the optimal power scheduling at the energy harvesting transmitters that
maximizes the sum-rate given harvested energy constraints. In [16], the authors have investigated joint
information and energy transfer in multi-cell cellular networks with single-antenna BSs and single-antenna
MSs. To the best of the authors’ knowledge, the general setup of multiple-input multiple-output (MIMO)
IFC models accounting for joint wireless information and energy transfer has not been addressed so far.
As an initial step, in this paper, we investigate a joint wireless information and energy transfer in a
two-user MIMO IFC, where each receiver either decodes the incoming information data (information
decoding, ID) or harvests the RF energy (energy harvesting, EH) to operate with a potentially perpetual
energy supply. Because practical circuits and hardware that harvest energy from the received RF signal
are not yet able to decode the information carried through the same RF signal [10], [11], [17], we assume
that the receiver cannot decode the information and simultaneously harvest energy. It is also assumed
that the two (Tx 1,Tx 2) transmitters have knowledge of their local CSI only, i.e. the CSI corresponding
to the links between a transmitter and all receivers (Rx 1, Rx 2). In addition, the transmitters do not
May 1, 2013
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3
share the information data to be transmitted and their CSI and, furthermore, the interference is assumed
not decodable at the receiver nodes as in [18]. That is, Tx 1 (Tx 2) cannot transfer the information
to Rx 2 (Rx 1). In a two-user IFC, we then have four different scenarios according to the Rx mode
– (ID1 , ID2 ), (EH1 , EH2 ), (EH1 , ID2 ), and (ID1 , EH2 ). Because, for (ID1 , ID2 ), the maximum
information bit rate is unknown and finding the optimal transmission strategy is still an open problem
in general, we investigate the achievable rate when a well-known iterative water-filling algorithm [19]–
[21] is adopted for (ID1 , ID2 ) with no CSI sharing between two transmitters. For (EH1 , EH2 ), we
derive the optimal transmission strategy achieving the maximum harvested energy. Because the receivers
operate in a single mode such as (ID1 , ID2 ) and (EH1 , EH2 ), when the information is transferred, no
energy is harvested from RF signals and vice versa. For (EH1 , ID2 ) and (ID1 , EH2 ), the achievable
energy-rate (R-E) trade-off region is not easily identified and the optimal transmission strategy is still
unknown. However, in this paper, we find a necessary condition of the optimal transmission strategy,
in which one of the transmitters should take a rank-one energy beamforming strategy with a proper
power control. Accordingly, the achievable R-E tradeoff region is identified for two different rank-one
beamforming strategies - maximum energy beamforming (MEB) and minimum leakage beamforming
(MLB). Furthermore, we also propose a new transmission strategy that satisfies the necessary condition signal-to-leakage-and-energy ratio (SLER) maximization beamforming. Note that the SLER maximizing
approach is comparable to the signal-to-leakage-and-noise ratio (SLNR) maximization beamforming [22],
[23] which has been developed for the multi-user MIMO data transmission, not considering the energy
transfer. The simulation results demonstrate that the proposed SLER maximization strategy exhibits wider
R-E region than the conventional transmission methods such as MLB, MEB, and SLNR beamforming.
Finally, we propose a mode scheduling method to switch between (EH1 , ID2 ) and (ID1 , EH2 ) based
on the SLER that further extends R-E tradeoff region.
The rest of this paper is organized as follows. In Section II, we introduce the system model for
two-user MIMO IFC. In Section III, we discuss the transmission strategy for two receivers on a single
mode, i.e. (ID1 , ID2 ) and (EH1 , EH2 ). In Section IV, we derive the necessary condition for the
optimal transmission strategy and investigate the achievable rate-energy (R-E) region for (EH1 , ID2 )
and (ID1 , EH2 ) and, in Section V, propose the SLER maximization strategy. In Section VI and Section
VII, we provide several discussion and simulation results, respectively, and in Section VIII we give our
conclusions.
Throughout the paper, matrices and vectors are represented by bold capital letters and bold lower-case
letters, respectively. The notations (A)H , (A)† , (A)i , [A]i , tr(A), and det(A) denote the conjugate
May 1, 2013
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4
Energy flow
Interference1
Receiver1( EH )
Transmitter1
Interference2
Transmitter2
Fig. 1.
Information flow
Receiver2 ( ID )
Two-user MIMO IFC in (EH1 , ID2 ) mode.
transpose, pseudo-inverse, the ith row, the ith column, the trace, and the determinant of a matrix A,
respectively. The matrix norm kAk and kAkF denote the 2-norm and Frobenius norm of a matrix A,
respectively, and the vector norm kak denotes the 2-norm of a vector a. In addition, (a)+ , max(a, 0)
and A 0 means that a matrix A is positive semi-definite. Finally, IM denotes the M × M identity
matrix.
II. S YSTEM
MODEL
We consider a two-user MIMO IFC system where two transmitters, each with Mt antennas, are
simultaneously transmitting their signals to two receivers, each with Mr antennas, as shown in Fig.
1. Note that each receiver can either decode the information or harvest energy from the received signal,
but it cannot execute the information decoding and energy harvesting at the same time due to the hardware
limitations. That is, each receiver can switch between ID mode and EH mode at each frame or time slot.1
We assume that the transmitters have perfect knowledge of the CSI of their associated links (i.e. the
links between a transmitter and all receivers) but do not share those CSI between them. In addition,
Mt = Mr = M for simplicity, but it can be extended to general antenna configurations. Assuming a
frequency flat fading channel, which is static over several frames, the received signal yi ∈ CM ×1 for
1
Note that the switching criterion between ID mode and EH mode depends on the receiver’s condition such as the available
energy in the storage and the required processing or circuit power. In this paper, we focus on the achievable rate and harvested
energy obtained by the transferred signals from both transmitters in the IFC according to the different receiver modes. The mode
switching policy based on the receiver’s condition is left as a future work. We assume that the mode decided by the receiver is
sent to both transmitters through the zero-delay and error-free feedback link at the beginning of the frame.
May 1, 2013
DRAFT
5
i = 1, 2 can be written as
y1 = H11 x1 + H12 x2 + n1 ,
y2 = H21 x1 + H22 x2 + n2 ,
(1)
where ni ∈ CM ×1 is a complex white Gaussian noise vector with a covariance matrix IM and Hij ∈
CM ×M is the normalized frequency-flat fading channel from the j th transmitter to the ith receiver such
P
(l,k) 2
(l,k)
as M
is the (l, k)th element of Hij and αij ∈ [0, 1]. We assume
l,k=1 |hij | = αij M [24]. Here, hij
that Hij has a full rank. The vector xj ∈ CM ×1 is the transmit signal, in which the independent messages
can be conveyed, at the j th transmitter with a transmit power constraint for j = 1 and 2 as
E[kxj k2 ] ≤ P for j = 1 and 2.
(2)
When the receiver operates in ID mode, the achievable rate at ith receiver, Ri , is given by [19]
−1
Ri = log det(IM + HH
ii R−i Hii Qi ),
(3)
where R−i indicates the covariance matrix of noise and interference at the ith receiver, i.e.,
R−1 = IM + H12 Q2 HH
12 ,
R−2 = IM + H21 Q1 HH
21 .
Here, Qj = E[xj xH
j ] denotes the covariance matrix of the transmit signal at the j th transmitter and,
from (2), tr(Qj ) ≤ P .
For EH mode, it can be assumed that the total harvested power Ei at the ith receiver (more exactly,
harvested energy normalized by the baseband symbol period) is given by
Ei = ζi E[kyi k2 ]


2
X
,
Hij Qj HH
= ζi tr 
ij + IM
(4)
j=1
where ζi denotes the efficiency constant for converting the harvested energy to electrical energy to be
stored [3], [10]. For simplicity, it is assumed that ζi = 1 and the noise power is negligible compared to
May 1, 2013
DRAFT
6
the transferred energy from each transmitters.2 That is,


2
X

Hij Qj HH
Ei ≈ tr 
ij
j=1
H
= tr Hi1 Q1 HH
i1 + tr Hi2 Q2 Hi2
= Ei1 + Ei2 ,
(5)
where Eij = tr Hij Qj HH
ij denoting the energy transferred from the j th transmitter to the ith receiver.
Interestingly, when the receiver decodes the information data from the associated transmitter under the
assumption that the signal from the other transmitter is not decodable [18], the signal from the other
transmitter becomes an interference to be defeated. In contrast, when the receiver harvests the energy, it
becomes a useful energy-transferring source. Fig. 1 illustrates an example of the receiving mode (EH1 ,
ID2 ), where the interference1 (dashed red line) should be reduced for ID, while the interference2 (dashed
green line) be maximized for EH. In what follows, for four possible receiving modes, we investigate the
achievable rate-harvested energy tradeoff. In addition, the corresponding transmission strategy (more
specifically, transmit signal design) is presented.
III. T WO
RECEIVERS ON A SINGLE MODE
A. Two IDs: maximum achievable sum rate
For the scenario (ID1 , ID2 ), it is desirable to obtain the maximum achievable sum rate. That is, the
problem can be formulated as follows:
(P 1) maximize
subject to
P2
i=1 Ri
tr(Qj ) ≤ P, Qj 0 for j = 1, 2,
(6)
(7)
The solution of (P1) has been extensively considered in many previous communication researches
[19]–[21], where the iterative water-filling algorithms have been developed to maximize the achievable
rate in a distributed manner with no CSI sharing between the transmitters. This is briefly summarized in
Algorithm 1:
Algorithm 1. Iterative Water-filling:
2
In this paper, we assume the system operates in the high signal-to-noise ratio (SNR) regime, which is also consistent with
the practical wireless energy transfer requires a high-power transmission, but we also discuss the low SNR regime in Section
VI as well.
May 1, 2013
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7
(0)
1) Initialize n = 0 and Qj ∈ QP for j = 1, 2, where
QP , {Q ∈ CM ×M : Q 0, tr(Q) = P }.
(8)
2) For n = 0 : Nmax , where Nmax is the maximum number of iterations3
(n+1)
for j = 1, 2 as follows:

 W F (H , R(n) , P ), if R(n) is updated,
jj
(n+1)
−j
−j
Qj
=
(n)

Q , otherwise,
Update Qj
(9)
j
(n)
where R−j indicates the covariance matrix of noise and interference in the j th receiver at the nth
iteration, i.e.,
(n)
(n)
(n)
(n)
R−1 = IM + H12 Q2 HH
12 ,
R−2 = IM + H21 Q1 HH
21 .
(n)
Note that R−j is measured at each receiver similarly to the way it has been done in [19] and,
(n)
furthermore, Qj
is computed at the receiver and reported to the transmitter through the zero-delay
and error-free feedback link.
max +1
3) Finally, Qj = QN
for j = 1, 2.
j
Here, W F () denotes the water-filling operator given as [19]:
+ H
W F (Hii , R, P ) = Ui (µi IM − D−1
i ) Ui ,
(10)
−1
H −1
where Ui and Di are obtained from the eigenvalue decomposition of HH
ii R Hii . That is, Hii R Hii =
Ui Di UH
i , and µi denotes the water level that satisfies the transmit power constraint as tr{(µi IM −
+
D−1
i ) } = P.
In the scenario (ID1 , ID2 ), because both receivers decode the information, the harvested energy
becomes zero.
B. Two EHs: maximum harvested energy
For the scenario (EH1 , EH2 ), both receivers want to achieve the maximum harvested energy. That is,
the problem can be formulated as:
(P 2) maximize
subject to
3
P2
i=1 Ei
tr(Qj ) ≤ P, Qj 0 for j = 1, 2,
(11)
(12)
Generally, Nmax = 20 is sufficient for the solutions to converge.
May 1, 2013
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8
The following proposition gives the optimal solution for the problem (P2).
H
Proposition 1: The optimal Qj for (P2) has a rank equal to one and
 is given
 as Qj = P [V̄j ]1 [V̄j ]1 ,
H1j
. That is, H̄j = Ūj Σ̄j V̄jH ,
where V̄j is a M ×M unitary matrix obtained from the SVD of H̄j , 
H2j
where Σ̄j = diag{σ̄j,1 , ..., σ̄j,M } with σ̄j,1 ≥ ... ≥ σ̄j,M .
Proof: From (5),
2
X
Ei =
2
X
i=1
i=1
=
2
X

tr 
2
X
j=1
2
X
tr


Hij Qj HH
ij
Hij Qj HH
ij
i=1
j=1
=
2
X
tr H̄j Qj H̄H
j
j=1
!
(13)
Note that the covariance matrix Qj can be written as Qj = Vj D2j VjH where Vj is a M × M unitary
P
2
m×n
matrix and D2j = diag{d2j,1 , ..., d2j,M } with M
m=1 dj,m ≤ P . Because tr(AB) = tr(BA) for A ∈ C
and B ∈ Cn×m , (13) can be rewritten as
2
X
Ei =
PM
2
m=1 dj,m
tr
D2j VjH H̄H
j H̄j Vj
j=1
i=1
Because
2
X
=
M
2 X
X
j=1 m=1
d2j,m kH̄j [Vj ]m k2 .
(14)
≤ P,
M
X
m=1
d2j,m kH̄j [Vj ]m k2 ≤ P max kH̄j [Vj ]m k2 .
(15)
m=1,...M
Here, the equality holds when d2j,m′ = P for m′ = arg max kH̄j [Vj ]m k2 and d2j,m = 0 for m 6= m′ ,
m=1,...M
which implies that Qj has a rank equal to one and accordingly, it is given as Qj = P [Vj ]m′ [Vj ]H
m′ .
Note that
2
kH̄j [Vj ]m′ k2 ≤ σ̄j,1
,
(16)
where the equality holds when [Vj ]m′ = [V̄j ]1 . Therefore, from (15) and (16), (14) is bounded as
2
X
i=1
Ei =
M
2 X
X
j=1 m=1
2
2
d2j,m kH̄j [V̄j ]m k2 ≤ P (σ̄1,1
+ σ̄2,1
),
and the equality holds when Qj = P [V̄j ]1 [V̄j ]H
1 .
Note that each transmitter can design the transmit covariance matrix Qj such that the transferred energy
from each transmitter is maximized without considering other transmitter’s channel information and
May 1, 2013
DRAFT
9
transmission strategy. That is, thanks to the energy conservation law, each transmitter transfers the energy
through its links independently.
From Proposition 1, the transmit signal on each transmitter can be designed as xj =
√
P [V̄j ]1 sj ,
where sj is any random signal with zero mean and unit variance. Because both receivers harvest the
energy and are not able to decode the information, the achievable rate becomes zero.
IV. O NE ID
RECEIVER AND
O NE EH
RECEIVER
In this section, without loss of generality, we will consider (EH1 , ID2 ) - the first receiver harvests
the energy and the second decodes information. The transmission strategy described below can also be
applied to (ID1 , EH2 ) without difficulty. Note that energy harvesting and information transfer occur
simultaneously in the IFC, and accordingly, the achievable rate-energy region is not trivial compared to
the scenarios (EH1 , EH2 ) and (ID1 , ID2 ).
A. A necessary condition for the optimal transmission strategy
Because information decoding is done only at the second receiver, by letting R = R2 and E = E1 =
E11 + E12 , we can define the achievable rate-energy region as:
(
−1
CR−E (P ) , (R, E) : R ≤ log det(IM + HH
22 R−2 H22 Q2 ),
E≤
)
P2
H
j=1 tr(H1j Qj H1j ), tr(Qj ) ≤ P, Qj 0, j = 1,2
.
(17)
Here, because EH and ID operations in the IFC interact with each other, the boundary of the rate-energy
region is not easily characterized and is so far unknown. The following lemma gives a useful insight into
the derivation of the optimal boundary.
Lemma 1: For H11 and H21 , there always exists an invertible matrix T ∈ CM ×M such that
UH
G H11 T = ΣG
H
VG
H21 T = IM ,
(18)
where UG and VG are unitary and ΣG is a diagonal matrix with σG,1 ≥ σG,2 ≥, ..., ≥ σG,M ≥ 0.
Proof: Because H21 has a full rank, by utilizing the generalized singular value decomposition [22],
[25], we can obtain an invertible matrix T′ such that
′
UH
G H11 T = ΣA
H
VG
H21 T′ = ΣB ,
May 1, 2013
DRAFT
10
where UG and VG are unitary and ΣA and ΣB are diagonal matrices with 1 ≥ σA,1 ≥ σA,2 ≥, ..., ≥
2 + σ 2 = 1. Therefore,
σA,M ≥ 0 and with 0 < σB,1 ≤ σB,2 ≤, ..., ≤ σB,M ≤ 1, respectively. Here, σA,i
B,i
−1
by setting T = T′ Σ−1
B , we can obtain (18) with ΣG = ΣA ΣB .
Without loss of generality, we set
Q1 = TXXH TH ,
(19)
where X ∈ CM ×m has the SVD as
X = Ux Σx VxH
with Σx = diag{σx,1 , ..., σx,m } and σx,1 ≥, ..., ≥ σx,m . Here,
m
X
2
= P ′,
σx,i
(20)
i=1
where P ′ is a normalization constant such that tr(TXXH TH ) ≤ P is satisfied. We then have the
following proposition.
Proposition 2: In the high SNR regime, the optimal Q1 at the boundary of the achievable rate-energy
region has a rank one at most. That is, rank(Q1) ≤ 1.
Proof: First, let us consider the boundary point (R̄, Ē ) of the achievable rate-energy, in which
Ē ≤ tr(H12 Q2 HH
12 ). Then, because the first transmitter do not need to transmit any signals causing the
interference to the ID receiver (the second receiver), Q1 = 0 is optimal. That is, rank(Q1) = 0.
For Ē > tr(H12 Q2 HH
12 ), let there be Q1 with m = rank(Q1 ) > 1 which corresponds to the boundary
point (R̄, Ē ) of the achievable rate-energy. Then, given the harvested energy Ē (the boundary point) and
Q2 , the covariance matrix Q1 exhibits
−1
R̄ = log det(IM + HH
22 R−2 H22 Q2 )
(21)
tr(H11 Q1 HH
11 ) = Ē11 ,
(22)
with
where Ē11 , Ē − tr(H12 Q2 HH
12 ). Because of Sylvester’s determinant theorem [26] (det(I + AB) =
det(I + BA) ), by substituting R−2 = IM + H21 Q1 HH
21 into (21), we can rewrite (21) as
H −1
R̄ = log det(IM + H22 Q2 HH
22 (IM + H21 Q1 H21 ) )
H −1
H −1
= log det((IM + H21 Q1 HH
+ H22 Q2 HH
21 )(IM + H21 Q1 H21 )
22 (IM + H21 Q1 H21 ) )
−1
H
H −1
= log det((IM + H21 Q1 HH
+ (H21 Q1 HH
21 )
21 + H22 Q2 H22 )(IM + H21 Q1 H21 ) )
H
H
= log det(IM + H21 Q1 HH
21 + H22 Q2 H22 ) − log det(IM + H21 Q1 H21 ).
May 1, 2013
(23)
DRAFT
11
Let us define m2 = rank(Q2 ) and consider m2 ≥ m without loss of generality. From Lemma 1 and
(19), (23) and (22) can be respectively rewritten as
H
H H
R̄ = log det(IM + VG XXH VG
+ H22 Q2 HH
22 ) − log det(IM + VG XX VG ),
H
H
= log det(IM + XXH + VG
H22 Q2 HH
22 VG ) − log det(IM + XX ),
(24)
and
H H H
H H H
tr(H11 Q1 HH
11 ) = tr(H11 TXX T H11 ) = tr(UG ΣG XX Σ UG )
= tr(ΣG XXH ΣG ) =
m
X
j=1
(i,j)
where ux
M
X
2
2
σG,i
|ux(i,j) |2 ) = Ē11 ,
σx,j
(
(25)
i=1
is the (i, j)th element of Ux . From the interlacing theorem (Theorem 3.1 in [27]), (24) can
be further rewritten as

m
m2
m
Y
Y
Y
2
2 
2
2

),
(1 + κj ) − log (1 + σx,i
(1 + σx,i + κi )
R̄ = log


≈ log 
i=1
j=m+1
i=1

m2
m
m
Y
Y
Y
2
2
2
2
),
κj − log (1 + σx,i
(σx,i + κi )
i=1
j=m+1
(26)
i=1
2
2
2
2
where κ2j is the interlaced value due to H22 Q2 HH
22 . That is, σy,m2 ≤ κj ≤ σy,1 , j = 1, ..., m2 , where σy,1
2
are the largest and the smallest eigenvalues of H22 Q2 HH
and σy,m
22 . Note that the last approximation
2
2
in (26) is from the high SNR regime (i.e., large power P such that log(1 + P ) ≈ log(P )), where σy,i
for all i = 1, ..., m2 are linearly proportional to P resulting in
κ2j ∝ P for j = 1, .., m2 .
(27)
Because σG,1 ≥, ..., ≥ σG,M ≥ 0 and
0 ≤ |ux(i,j) |2 ≤ 1,
May 1, 2013
M
X
i=1
|ux(i,j) |2 = 1,
DRAFT
12
if there exists m > 1 such that (25) is satisfied with (20), we can find Q′1 with rank(Q′1 ) = 1 satisfying
(25). In addition, (26) can be rewritten as:
Q
R̄ ≈ log 

= log 

≈ log 
= log

m
m2
2 + κ2 ) Y
(σx,i
i
Qi=1
κ2j 
m
2 )
(1
+
σ
i=1
x,i j=m+1
m
Y
2
σx,i
2
1 + σx,i
i=1
m
Y
κ2i
2
1 + σx,i
i=1
+
!
Qm2
κ2
Qm i=1 i 2
i=1 (1 + σx,i )
κ2i
2
1 + σx,i
m2
Y
j=m+1
!
!
m2
Y
j=m+1


κ2j 
κ2j 
(28)
(29)
.
(30)
2 is negligible with respect
The approximation in (29) is from (27) with a large P . That is because σx,i
Q
2
to κ2i when Ē is finite. From (20), m
i=1 (1 + σx,i ) in the denominator of (30) has the minimum value
when m = 1. In other words, if Q1 with m > 1 exhibits (R̄, Ē ), then we can find Q′1 with m = 1 such
that (R̄′ , Ē ) with R̄′ > R̄ in the high SNR regime, which contradicts that the point (R̄, Ē ) is a boundary
point.
2
Remark 1: Note that when the required harvested energy Ē (more precisely, Ē11 ) is large, both σx,i
2 are linearly proportional to P resulting in
and σy,i
2
σx,i
, κ2j ∝ P for i = 1, ..., m, j = 1, .., m2 .
(31)
Then, (28) becomes:

Therefore,

(32)
R̄ ∝ log P m2 −m = (m2 − m) log P,
(33)
R̄ ≈ log 
m
Y
i=1
1+
κ2i
2
σx,i
!
m2
Y
j=m+1
κ2j  .
which implies that in the high SNR regime with large harvesting energy E , the achievable rate is linearly
proportional to (m2 − m). Then, we can easily find that it is maximized when m = 1. Note that it can
be interpreted as the degree of freedom (DOF) in the IFC [28], in which by reducing the rank of the
transmit signal at the first transmitter, the DOF at the second transceiver can be increased.
Remark 2: Intuitively, from the power transfer point of view, Q1 should be as close to the dominant
eigenvector of HH
11 H11 as possible, which implies that the rank one is optimal for power transfer. From
the information transfer point of view, when SNR goes to infinity, the rate maximization is equivalent to
May 1, 2013
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13
the DOF maximization. That is, a larger rank for Q1 means that more dimensions at the second receiver
will be interfered. Therefore, a rank one for Q1 is optimal for both information and power transfer.
When each node has a single antenna (M = 1), the scalar weight at the j th transmitter can be written
p
as Pj ejθj or simply, Qj = Pj . The achievable rate-energy region can then be given as
(
CR−E (P ) , (R, E) : R ≤ log(1 +
P2 |h22 |2
1+P1 |h21 |2 ),
)
E ≤ P1 |h11 |2 + P2 |h12 |2 , Pj ≤ P, j = 1,2 .
(34)
From (34), we can easily find that P2 = P at the boundary of the achievable rate-energy region. That is,
the second transmitter always transmits its signal with full power P . Therefore, the optimal transmission
strategy for M = 1 boils down to the power allocation problem of the first transmitter in the IFC.
From Proposition 2, when transferring the energy in the IFC, the transmitter’s optimal strategy is
either a rank-one beamforming or no transmission according to the energy transferred from the other
transmitter, which increases the harvested energy at the corresponding EH receiver and simultaneously
reduce the interference at the other ID receiver. Even though the identification of the optimal achievable
R-E boundary is an open problem, it can be found that the first transmitter will opt for a rank-one
beamforming scheme. Therefore, in what follows, we first design two different rank-one beamforming
schemes for the first transmitter and identify the achievable rate-energy trade-off curves for the two-user
MIMO IFC where the rank-one beamforming schemes are exploited.
B. Rank-one Beamforming Design
1) Maximum-energy beamforming (MEB): Because the first receiver operates as an energy harvester,
the first transmitter may steer its signal to maximize the energy transferred to the first receiver, resulting
in a considerable interference to the second receiver operating as an information decoder.
From Proposition 2, the corresponding transmit covariance matrix Q1 is then given by
Q1 = P1 [V11 ]1 [V11 ]H
1 ,
(35)
where V11 is a M × M unitary matrix obtained from the SVD of H11 and 0 ≤ P1 ≤ P . That is,
H , where Σ
H11 = U11 Σ11 V11
11 = diag{σ11,1 , ..., σ11,M } with σ11,1 ≥ ... ≥ σ11,M . Here, the energy
2 .
harvested from the first transmitter is given by P1 σ11,1
May 1, 2013
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14
2) Minimum-leakage beamforming (MLB): From an ID perspective at the second receiver, the first
transmitter should steer its signal to minimize the interference power to the second receiver. That is, from
Proposition 2, the corresponding transmit covariance matrix Q1 is then given by
Q1 = P1 [V21 ]M [V21 ]H
M,
(36)
where V21 is a M × M unitary matrix obtained from the SVD of H21 and 0 ≤ P1 ≤ P . That is,
H , where Σ
H21 = U21 Σ21 V21
21 = diag{σ21,1 , ..., σ21,M } with σ21,1 ≥ ... ≥ σ21,M . Then, the energy
harvested from the first transmitter is given by P1 kH11 [V21 ]M k2 .
C. Achievable R-E region
Given Q1 as in either (35) or (36), the achievable rate-energy region is then given as:
(
CR−E (P ) =
(R, E) : R = R2 , E = E11 + E12 ,
−1
H
R2 ≤ log det(IM + HH
22 R−2 H22 Q2 ), E12 ≤ tr(H12 Q2 H12 ),
)
tr(Q2 ) ≤ P, Q2 0 ,
(37)
where
E11 =
and
R−2


2
P1 σ11,1
for MEB
 P kH [V ] k2
1
11
21 M
,
(38)
for MLB

 I + P H [V ] [V ]H HH for MEB
M
1 21
11 1
11 1
21
=
.
 I + P σ 2 [U ] [U ]H for MLB
M
1 21,M
21 M
21 M
(39)
2
Note that because σ11,1
≥ kH11 [V21 ]M k2 , the energy harvested by the first receiver from the first
transmitter with MEB is generally larger than that with MLB.
Due to Sylvester’s determinant theorem, R2 can be derived as:
−1
R2 = log det(IM + HH
22 R−2 H22 Q2 )
−1/2
−1/2
= log det(IM + R−2 H22 Q2 HH
22 R−2 ).
(40)
−1/2
Accordingly, by letting H̃22 = R−2 H22 , we have the following optimization problem for the rate-energy
May 1, 2013
DRAFT
15
region of (37)4
(P 3) maximize
Q2
log det(IM + H̃22 Q2 H̃H
22 )
(41)
subject to tr(H12 Q2 HH
12 ) ≥ max(Ē −E11 , 0)
tr(Q2 ) ≤ P, Q2 0,
(42)
(43)
where Ē can take any value less than Emax denoting the maximum energy transferred from both
transmitters. Here, it can be easily derived that Emax is given as

2 ) for MEB
2

+ σ12,1
P (σ11,1
,
Emax =
 P (kH [V ] k2 + σ 2 ) for MLB
11
21 M
12,1
(44)
2
is the largest singular value of H12 and it is achieved when the second transmitter also steers
where σ12,1
its signal such that its beamforming energy is maximized on the cross-link channel H12 . That is,
Q2 = P [V12 ]1 [V12 ]H
1 .
Note that the corresponding transmit signal is given by x2 (n) =
(45)
√
P [V12 ]1 s2 (n), where s2 (n) is a random
signal with zero mean and unit variance. Therefore, when s2 (n) is Gaussian randomly distributed with
zero mean and unit variance, which can be realized by using a Gaussian random code [30], the achievable
H
rate is given by R2 = log det(IM + P H̃22 [V12 ]1 [V12 ]H
1 H̃22 ).
Note that because E11 in (42) and H̃22 in (41) are dependent on P1 (≤ P ), we identify the achievable
R-E region iteratively as:
Algorithm 2. Identification of the achievable R-E region:
(0)
1) Initialize n = 0, P1
= P,
(0)
E11 =
and
(0)
R−2
4


(0)
2
P1 σ11,1
for MEB
 P (0) kH [V ] k2
11
21 M
1
,
(46)
for MLB

 I + P (0) H [V ] [V ]H HH for MEB
M
21
11 1
11 1
21
1
=
.
(0)
 I + P σ 2 [U ] [U ]H for MLB
M
21
M
21
1
21,M
M
(47)
The dual problem of maximizing energy subject to rate constraint can be formulated, but the rate maximization problem is
preferred because it can be solved using approaches similar to those in the rate maximization problems under various constraints
[10], [19], [29].
May 1, 2013
DRAFT
16
2) For n = 0 : Nmax
(n)
Solve the optimization problem (P3) for Q2
(n)
(n)
If tr(H12 Q2 HH
12 ) + E11 > Ē
(n+1)
P1
where κ =


for MEB
 kH [V ] k2
11
21 M
(n+1)
Then, P1
2
σ11,1
= max
(n+1)
= min(P, P1
(n)
(n)
as a function of E11 and R−2 .
!
(n)
)
Ē − tr(H12 Q2 HH
12
,0 ,
κ
(48)
.
for MLB
(n+1)
) and update E11
(n+1)
and R−2
(n+1)
with P1
similarly to (46) and
(47).
3) Finally, the boundary point of the achievable R-E region is given as (R, E) = (log det(IM +
(Nmax +1)
H̃22 Q2
(N
max
H̃H
22 ), E11
+1)
(Nmax +1)
+ tr(H12 Q2
HH
12 )).
(n)
(n)
In (48), if the total transferred energy (tr(H12 Q2 HH
12 ) + E11 ) is larger than the required harvested
energy Ē , the first transmitter reduces the transmit power P1 to lower the interference to the ID receiver.
(n)
In addition, if the energy harvested by the first receiver from the second transmitter (tr(H12 Q2 HH
12 ))
is larger than Ē , the first transmitter does not transmit any signal. That is, rank(Q) = 0 as claimed in
the proof of Proposition 2.
(n)
To complete Algorithm 2, we now show how to solve the optimization problem (P3) for Q2
(n)
in Step
(n)
2 of Algorithm 2. The optimization problem (P3) with E11 and R−2 can be tackled with two different
approaches according to the value of Ē , i.e., 0 ≤ Ē ≤ E11 and E11 < Ē ≤ Emax . Note that we have
dropped the superscript of the iteration index (n) for notation simplicity. For 0 ≤ Ē ≤ E11 , (P3) becomes
the conventional rate maximization problem for single-user effective MIMO channel (i.e., H̃22 ) whose
solution is given as
Q2 = W F (H̃22 , IM , P ),
(49)
resulting in the maximum achievable rate for the given rank-one strategy Q1 . Here, the operator W F ()
is defined in (10).
For E11 < Ē ≤ Emax , the optimization problem (P3) can be solved by a “water-filling-like” approach
similar to the one appeared in the joint wireless information and energy transmission optimization with
a single transmitter [10]. That is, the Lagrangian function of (P 3) can be written as
L(Q2 , λ, µ) = log det(IM + H̃22 Q2 H̃H
22 )
+λ(tr(H12 Q2 HH
12 ) − (Ē −E1 )) − µ(tr(Q2 ) − P ),
May 1, 2013
DRAFT
17
and the corresponding dual function is then given by [10], [29]
g(λ, µ) = max L(Q2 , λ, µ).
(50)
Q2 0
Here the optimal solution µ′ , λ′ , and Q2 can be found through the iteration of the following steps [29]
1) The maximization of L(Q2 , λ, µ) over Q2 for given λ, µ.
2) The minimization of g(λ, µ) over λ, µ for given Q2 .
Note that, for given λ, µ, the maximization of L(Q2 , λ, µ) can be simplified as
max L(Q2 , λ, µ) = log det(IM + H̃22 Q2 H̃H
22 ) − tr(AQ2 ),
(51)
Q2 0
where A = µIM − λHH
12 H12 . Note that (51) is the point-to-point MIMO capacity optimization with a
single weighted power constraint and the solution is then given by [10], [29]
′
′
′H −1/2
Q2 = A−1/2 Ṽ22
Λ̃ Ṽ22
A
,
(52)
′
′ is obtained from the SVD of the matrix H̃ A−1/2 , i.e., H̃ A−1/2 = Ũ′ Σ̃ Ṽ′H . Here,
where Ṽ22
22
22
22 22 22
′
′
′
′
′
′
Σ̃22 = diag{σ̃22,1
, ..., σ̃22,M
} with σ̃22,1
≥ ... ≥ σ̃22,M
≥ 0 and Λ̃ = diag{p̃1 , ..., p̃M } with p̃i =
′2 )+ , i = 1, ..., M . The parameters µ and λ minimizing g(λ, µ) in Step 2 can be solved by the
(1 − 1/σ̃22,i
subgradient-based method [10], [31], where the the subgradient of g(λ, µ) is given by (tr(H12 Q2 HH
12 ) −
(Ē −E1 ), P − tr(Q2 )).
V. E NERGY- REGULARIZED SLER- MAXIMIZING
BEAMFORMING
In Section IV-B, two rank-one beamforming strategies are developed according to different aims either maximizing transferred energy to EH or minimizing interference (or, leakage) to ID. Note that
in [22], [23], the maximization of the ratio of the desired signal power to leakage of the desired signal
on other users plus noise measured at the transmitter, i.e., SLNR maximization, has been utilized in the
beamforming design in the multi-user MIMO system. Similarly, in this section, to maximize transferred
energy to EH and simultaneously minimize the leakage to ID, we define a new performance metric,
signal-to-leakage-and-harvested energy ratio (SLER) as
SLER =
kH11 vk2
.
kH21 vk2 + max(Ē − P1 kH11 k2 , 0)
(53)
Note that the noise power contributes to the denominator of SLNR in the beamforming design [22],
[23] because the noise at the receiver affects the detection performance degradation for information
transfer. That is, the noise power should be considered in the computation of beamforming weights. In
contrast, the contribution of the minimum required harvested energy is added in SLER of (53), because
May 1, 2013
DRAFT
18
the required harvested energy minus the energy directly harvested from the first transmitter is a main
performance barrier of the EH receiver. Therefore, in the energy beamforming, the required harvested
energy is considered in the computation of the beamforming weights.5 Then, the SLER of (53) can be
rewritten as
SLER =
vH
HH
21 H21
v H HH
11 H11 v
.
+ max(Ē/P1 − kH11 k2 , 0)IM v
The beamforming vector v that maximizes SLER of (54) is then given by
√
P1
v=
v̄,
kv̄k
(54)
(55)
where v̄ is the generalized eigenvector associated with the largest generalized eigenvalue of the matrix
2
H
pair (HH
11 H11 , H21 H21 + max(Ē/P1 − kH11 k , 0)IM ). Here, v̄ can be efficiently computed by using a
generalized singular value decomposition (GSVD) algorithm [23], [32], which is briefly summarized in
Algorithm 3.
Algorithm 3. SLER maximizing GSVD-based beamforming:


H11




1) Set K = 
 ∈ C3M ×M .
H21

 q
2
max Ē/P1 − kH11 k , 0 IM
2) Compute QR decomposition (QRD) of K = [Pα ; Pβ ] R̄ , where [Pα ; Pβ ] is unitary and R̄ ∈
CM ×M is upper triangular. Here, Pα ∈ C2M ×M .
3) Compute V̄α from the SVD of Pα , i.e., ŪH
α (Pα )1:M V̄α = Σ̄α .
4) v̄ = R̄−1 [V̄α ]1 and then, v =
√
P1
kv̄k v̄.
Here, because



K=
q
as in [32], for P1 kH11 k2 < Ē
H21
max Ē/P1 − kH11 k2 , 0 IM
R̄−1 = q
5

H11


 = [Pα ; Pβ ] R̄

1
Pβ ,
max Ē/P1 − kH11 k2 , 0
Strictly speaking, the SLER can be defined as SLER =
kH11 vk2
.
kH21 vk2 +max(Ē−kH11 vk2 ,0)
(56)
However, for computational simplicity,
the lower bound on the required harvested energy is added in the denominator of SLER from the fact that kH11 vk2 ≤ P1 kH11 k2 .
May 1, 2013
DRAFT
19
which avoids a matrix inversion in Step 4 of Algorithm 3. Because Algorithm 3 requires one QRD of an
3M × M matrix (Step 2), one SVD of an M × M matrix (Step3), and one (M × M , M × 1) matrix-vector
multiplication (Step 4 with (56)), it has a slightly more computational complexity compared to the MEB
and the MLB in Section IV-B that need one SVD of an M × M matrix.
Once the beamforming vector is given as (55), we can obtain the R-E tradeoff curve for SLER
maximization beamforming by taking the approach described in Section IV-C. Interestingly, from (54),
when the required harvested energy at the EH receiver is large, the matrix in the denominator of (53)
approaches an identity matrix multiplied by a scalar. Accordingly, the SLER maximizing beamforming is
√
equivalent with the MEB in Section IV-B.1. That is, v becomes P1 [V11 ]1 . In contrast, as the required
harvested energy becomes smaller, v is steered such that less interference is leaked into the ID receiver
to reduce the denominator of (53). That is, v approaches the MLB weight vector in Section IV-B.2.
Therefore, the proposed SLER maximizing beamforming weighs up both metrics - energy maximization
to EH and leakage minimization to ID.
Note that the SLER value indicates how suitable a receiving mode, (EH1 , ID2 ) or (ID1 , EH2 ), is
to the current channel. This motivates us to propose a mode scheduling between (EH1 , ID2 ) and (ID1 ,
EH2 ). That is, higher SLER implies that the transmitter can transfer more energy to its associated EH
receiver incurring less interference to the ID receiver. Based on this observation, our scheduling process
can start with evaluating for a given interference channel and P ,
kH11 vk2
kH21 vk2 + max(Ē − P kH11 k2 , 0)
(57)
kH22 vk2
.
kH12 vk2 + max(Ē − P kH22 k2 , 0)
(58)
SLER(1) = max
v
and
SLER(2) = max
v
If SLER(1) ≥ SLER(2) , (EH1 , ID2 ) is selected. Otherwise, (ID1 , EH2 ) is selected.
VI. D ISCUSSION
A. The rank-one optimality in the low SNR regime for one ID receiver and one EH receiver
Even though we have assumed the high SNR regime throughout the paper, in some applications such as
wireless ad-hoc sensor networks, low power transmissions are also considered. The following proposition
establishes the rank-one optimality in the low SNR regime.
Proposition 3: Considering (EH1 , ID2 ) without loss of generality, in the low SNR regime, the optimal
Q1 at the boundary of the achievable rate-energy region has a rank one.
May 1, 2013
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20
Proof: Similarly to (23), the achievable rate at the ID2 receiver is given by
H
H
R̄ = log det(IM + H21 Q1 HH
21 + H22 Q2 H22 ) − log det(IM + H21 Q1 H21 ).
(59)
For a Hermitian matrix A with eigenvalues in (−1, 1), log det(I + A) can be extended as [33]
1
1
log det(I + A) = tr(A) − tr(A2 ) + tr(A3 ) + ....
2
3
(60)
Because Hij Qj HH
ij is Hermitian and positive definite, and its maximum eigenvalue is upper-bounded
H
as λmax (Hij Qj HH
ij ) < λmax (Qj )λmax (Hij Hij ) [33], for sufficiently low transmission power, their
maximum eigenvalues lie in (−1, 1). Accordingly, in the low SNR regime, R̄ can be approximated as
H
H
R̄ ≈ tr(H21 Q1 HH
21 + H22 Q2 H22 ) − tr(H21 Q1 H21 )
= tr(H22 Q2 HH
22 ).
(61)
That is, the achievable rate is independent of the interference from the first transmitter (noise-limited
system). Then, Q1 at the first transmitter can be designed to maximize the harvested energy. Therefore,
the optimal Q1 at the boundary of the achievable rate-energy region is given by
Q1 = arg max tr(H11 QH11 ).
Q
(62)
2 , where the equality is satisfied when Q = P [V ] [V ]H as in (35).
Note that tr(H11 QH11 ) ≤ P σ11,1
11 1
11 1
Therefore, the optimal Q1 at the boundary of the achievable rate-energy region has a rank one.
Note that, from (61), Q2 maximizing R̄ is designed as
Q2 = arg max tr(H22 QHH
22 )
Q
(63)
and the corresponding Q2 is given by Q2 = P [V22 ]1 [V22 ]H
1 , where V22 is the right singular matrix of
H22 . That is, at the low SNR, the optimal information transfer strategy in the joint information and energy
transfer system is also a rank-one beamforming, which is consistent with the result in the information
transfer system [34], where the region that the beamforming is optimal becomes broader as the SNR
decreases.
B. Asymptotic behavior for a large M
Note that Proposition 2 gives us an insight on the joint information and energy transfer with a large
number of antennas describing a promising future wireless communication structure such as a massive
MIMO system [35]–[37].
May 1, 2013
DRAFT
21
Given the normalized channel H =
√
M
H̃,
kH̃k
where the elements of H̃ are i.i.d. zero-mean complex
Gaussian random variables (RVs) with a unit variance, and Q = P vvH with a finite P and kvk = 1,
we define R = IM + HQHH which is analogous to R−2 in (21) of the proof of Proposition 2. Then,
because det(R) = 1 + P vH HH Hv and
HH H ≈
1 H
H̃ H̃ ≈ IM ,
M
(Central limit theorem in [38])
when M goes to infinity, det(R) ≈ 1 + P and it is independent from the beamforming vector v.
Analogously, when M increases, the design of v1 in Q1 = P1 v1 v1H at the first transmitter is independent
from det(R−2 ) (accordingly, independent from H21 ). Therefore, when nodes have a large number of
antennas, the transmit signal for energy transfer can be designed by caring about its own link, not caring
about the interference link to the ID receiver. That is, for a large M , MEB with a power control becomes
optimal because it maximizes the energy transferred to its own link.
Remark 3: Interestingly, from Section VI-A and VI-B we note that, when the SNR decreases or the
number of antennas increases, the energy transfer strategy in the MIMO IFC would be designed by only
caring about its own link to the EH receiver, not by considering the interference or leakage through
the other link to the ID receiver. In addition, massive MIMO effect makes the joint information and
energy transfer in the MIMO IFC naturally split into disjoint information and energy transfer in two
non-interfering links.
VII. S IMULATION R ESULTS
Computer simulations have been performed to evaluate the R-E tradeoff of various transmission
strategies in the two-user MIMO IFC. In the simulations, the normalized channel Hij is generated
√
αij M
such as Hij = kH̃ k H̃ij , where the elements of H̃ij are independent and identically distributed (i.i.d.)
ij
F
zero-mean complex Gaussian random variables (RVs) with a unit variance. The maximum transmit power
is set as P = 50W , unless otherwise stated.
Table I lists the achievable rate and energy, (R, E ), for single modes – (EH1 , EH2 ) and (ID1 , ID2 ).
The harvested energy of (EH1 , EH2 ) for M = 4 is larger than that for M = 2 and furthermore, the
achievable rate of (ID1 , ID2 ) for M = 4 is higher than that for M = 2. Note that the achievable rate
of (EH1 , EH2 ) and the harvested energy of (ID1 , ID2 ) are zero.
Fig. 2 shows R-E tradeoff curves for the MEB and the MLB described in Section IV-B when M = 4,
αii = 1 for i = 1, 2, and αij = 0.8 for i 6= j . The first transmitter takes a rank-one beamforming, either
MEB or MLB, and the second transmitter designs its transmit signal as (45), (49), and (52), described in
May 1, 2013
DRAFT
22
TABLE I
T HE ACHIEVABLE RATE AND ENERGY, (R, E), FOR SINGLE MODES WHEN M ∈ {2, 4}
Mode
M =2
M =4
(EH1 , EH2 )
(0, 262.98)
(0, 359.57)
(ID1 , ID2 )
(9.67, 0)
(16.08, 0)
Section IV-C. As expected, the MEB strategy raises the harvested energy at the EH receiver, while the
MLB increases the achievable rate at the ID receiver. Interestingly, in the regions where the energy is less
than a certain threshold around 45 Joule/sec, the first transmitter does not transmit any signals to reduce
the interference to the second ID receiver. That is, the energy transferred from the second transmitter is
sufficient to satisfy the energy constraint at the EH receiver.
The dashed lines indicate the R-E curves of the time-sharing of the full-power rank-one beamforming
(either MEB or MLB) and the no transmission at the first transmitter. Here, the second transmitter
switches between the beamforming on H12 as (45) and the water-filling as (49) in the corresponding
time slots. For MLB, “water-filling-like” approach (52) exhibits higher R-E performance than the timesharing scheme. However, for MEB, when the energy is less than 120 Joule/sec, the time-sharing exhibits
better performance than the approach (52). That is, because the MEB causes large interference to the ID
receiver, it is desirable that, for the low required harvested energy, the first transmitter turns off its power
in the time slots where the second transmitter is assigned to exploit the water-filling method as (49).
Instead, in the remaining time slots, the first transmitter opts for a MEB with full power and the second
transmitter transfers its information to the ID receiver by steering its beam on EH receiver’s channel
H12 as (45) to help the EH operation. In Fig. 3, we have additionally included the R-E tradeoff curves
for MEB with rank(Q1) = 2 when the simulation parameters are the same as those of Fig. 2. Here,
we can find that the MEB with rank(Q1) = 1 has superior R-E boundary points compared to that with
rank(Q1 ) = 2. That is, even though we have not identified the exact optimal R-E boundary, for a given
beamforming (MEB in Fig. 3), we can find that the beamforming with rank(Q1) = 1 has superior R-E
boundary points compared to that with rank(Q1 ) = 2.
In Fig. 4, we plot R-E tradeoff curves for M = 4 and P = 0.1. As observed in Section VI-A, at
the low SNR, the MEB exhibits higher harvested energy than the MLB without any degradations in the
achievable rate. Fig. 5 shows R-E tradeoff curves for M = 15. Compared to M ∈ 4 (Fig. 2), the gap
May 1, 2013
DRAFT
23
R−E tradeoff curves when Mr=Mt=4
250
Max. energy beamforming
Min. leakage beamforming
Time−sharing of full/zero
power transmissions at 1st Tx.
Energy (Joule/sec)
200
150
100
rank(Q1)=1
50
No transmission at 1st Tx.
rank(Q1)=0
0
Fig. 2.
9.5
10
10.5
11
11.5
Rate (bits/s/Hz)
12
12.5
13
13.5
R-E tradeoff curves for MEB and MLB when M = 4, αii = 1 for i = 1, 2, and αij = 0.8 for i 6= j.
between the achievable rates of MEB and MLB is relatively less apparent. As pointed out in Remark 3
of Section VI, for low SNRs or large numbers of antennas in the MIMO IFC, the energy transfer strategy
of maximizing the transferred energy on its own link exhibits wider R-E region than that of minimizing
the interference to the other ID receiver.
Fig. 6 shows R-E tradeoff curves for MEB, MLB, SLNR maximizing beamforming, and SLER
maximizing beamforming when M = 4, αii = 1 for i = 1, 2, and αij = 0.8 for i 6= j . The R-E
region of the proposed SLER maximizing beamforming covers most of those of both MEB and MLB,
while the SLNR beamforming does not cover the region for MEB. Fig. 7 shows R-E tradeoff curves for
an asymmetric case Mt = 3 and Mr = 4. We can find a similar trend with Mt = 4 = Mr = 4, but the
overall harvested energy with Mt = 3 and Mr = 4 is slightly less than that with Mt = 4 = Mr = 4.
Fig. 8 shows the R-E tradeoff curves for SLER maximizing beamforming with/without SLER-based
scheduling described in Section V when (a) αij = 0.7 and (b) αij = 1 for i 6= j . Here, we set αii = 1
for i = 1, 2 and M = 2. Note that the case with αij = 0.7 has weaker cross-link channel (inducing less
interference) than that with αij = 1. The SLER-based scheduling extends the achievable R-E region for
May 1, 2013
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24
R−E tradeoff curves when Mr=Mt=4
250
Max. energy beamforming with rank(Q1)=1
Min. leakage beamforming with rank(Q1)=1
Max. energy beamforming with rank(Q1)=2
200
Energy (Joule/sec)
Time−sharing of full/zero
power transmissions at 1st Tx.
150
100
50
No transmission at 1st Tx.
rank(Q1)=0
0
Fig. 3.
8
9
10
11
Rate (bits/s/Hz)
12
13
R-E tradeoff curves for MEBs (rank(Q1 ) = {1, 2}) and MLB (rank(Q1 ) = 1) when M = 4, αii = 1 for i = 1, 2,
and αij = 0.8 for i 6= j.
both αij ∈ {0.7, 1}, but the improvement for αij = 1 is slightly more apparent. That is, the SLER-based
scheduling becomes more effective when strong interference exists in the system. Note that the case with
αij = 1 exhibits slightly lower achievable rate than that with αij = 0.7, while achieving larger harvested
energy. That is, the strong interference degrades the information decoding performance but it can be
effectively utilized in the energy-harvesting.
VIII. C ONCLUSION
In this paper, we have investigated the joint wireless information and energy transfer in two-user
MIMO IFC. Based on Rx mode, we have different transmission strategies. For single-operation modes
- (ID1 , ID2 ) and (EH1 , EH2 ), the iterative water-filling and the energy-maximizing beamforming on
both receivers can be adopted to maximize the information bit rate and the harvested energy, respectively.
For (EH1 , ID2 ), and (ID1 , EH2 ), we have found a necessary condition of the optimal transmission
strategy that one of transmitters should take a rank-one beamforming with a power control. Accordingly,
for two transmission strategies that satisfy the necessary condition - MEB and MLB, we have identified
May 1, 2013
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25
R−E tradeoff curves when M =M =4 and P=0.1
r
t
0.5
Max. energy beamforming
Min. leakage beamforming
0.45
0.4
Energy (Joule/sec)
0.35
0.3
0.25
0.2
0.15
0.1
0.23
Fig. 4.
0.24
0.25
0.26
0.27
0.28
Rate (bits/s/Hz)
0.29
0.3
0.31
0.32
R-E tradeoff curves for MEB and MLB when M = 4 and P = 0.1.
their achievable R-E tradeoff regions, where the MEB (MLB) exhibits larger harvested energy (achievable
rate). We have also found that when the SNR decreases or the number of antennas increases, the joint
information and energy transfer in the MIMO IFC can be naturally split into disjoint information and
energy transfer in two non-interfering links. Finally, we have proposed a new transmission strategy satisfying the necessary condition - signal-to-leakage-and-energy ratio (SLER) maximization beamforming
which shows wider R-E region than the conventional transmission methods. That is, we have found that
even though the interference degrades the ID performance in the two-user MIMO IFC, the proposed
SLER maximization beamforming scheme effectively utilizes it in the EH without compromising ID
performance.
Note that, motivated from the rank-one beamforming optimality, the identification of the optimal R-E
boundary will be a challenging future work. Furthermore, the partial CSI or erroneous channel information
degrades the achievable rate and the harvested energy at the receivers, which drives us to develop a robust
rank-one beamforming.
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26
R−E tradeoff curves when Mr=Mt=15
350
Max. energy beamforming
Min. leakage beamforming
300
Energy (Joule/sec)
250
200
150
100
rank(Q1)=1
50
No transmission at 1st Tx.
rank(Q1)=0
0
15
20
25
30
Rate (bits/s/Hz)
Fig. 5.
R-E tradeoff curves for MEB and MLB when M = 15.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers whose comments helped us improve this
paper.
R EFERENCES
[1] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless power transfer via strongly
coupled magnetic resonances,” Science, vol. 137, no. 83, pp. 83–86, Sept. 2007.
[2] M. Pinuela, D. Yates, S. Lucyszyn, and P. Mitcheson, “Maximizing DC-load efficiency for semi-resonant inductive power
transfer,” submitted to IEEE Transactions on power electronics, 2012.
[3] R. J. M. Vullers, R. V. Schaijk, I. Doms, C. V. Hoof, and R. Merterns, “Micropower energy harvesting,” Solid-State
Electronics, vol. 53, no. 7, pp. 684–693, July 2009.
[4] T. Le, K. Mayaram, and T. Fiez, “Efficient far-field radio frequency energy harvesting for passively powered sensor
networks,” IEEE J. Solid-State Circuits, vol. 43, no. 5, pp. 1287–1302, May 2008.
[5] O. Ozel, K. Tutuncuoglu, J. Yang, S. Ulukus, and A. Yener, “Transmission with energy harvesting nodes in fading wireless
channels: Optimal policies,” IEEE J. Select. Areas Commun., vol. 29, no. 8, pp. 1732–1743, Sept. 2011.
[6] R. Rajesh, V. Sharma, and P. Viswanath, “Information capacity of energy harvesting sensor nodes,” in Proc. IEEE
International Symposium on Information Theory, 2011, July 2011, pp. 2363–2367.
May 1, 2013
DRAFT
27
R−E tradeoff curves when Mr=Mt=4
250
Max. energy beamforming
Min. leakage beamforming
SLNR max. beamforming [22], [23]
SLER max. beamforming
Energy (Joule/sec)
200
150
100
rank(Q1)=1
50
No transmission at 1st Tx.
rank(Q1)=0
0
Fig. 6.
9
9.5
10
10.5
11
Rate (bits/s/Hz)
11.5
12
12.5
13
R-E tradeoff curves for MEB, the MLB, SLNR maximizing beamforming, and SLER maximizing beamforming when
M = 4, αii = 1 for i = 1, 2, and αij = 0.8 for i 6= j.
[7] ——, “Information capacity of an energy harvesting sensor node,” submitted to IEEE Transactions on Information Theory,
http://arxiv.org/abs/1212.3177, 2012.
[8] K. Ishibashi, H. Ochiai, and V. Tarokh, “Energy harvesting cooperative communications,” in Proc. IEEE International
Symposium on Personal, Indoor and Mobile Radio Communications, 2012, Sept. 2012, pp. 1819–1823.
[9] Y. Luo, J. Zhang, and K. B. Letaief, “Optimal scheduling and power allocation for two-hop energy harvesting
communication systems,” submitted to IEEE Transactions on Wireless Communications, http://arxiv.org/abs/1212.5394,
2012.
[10] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” submitted to
IEEE Transactions on Wireless Communications, http://arxiv.org/abs/1105.4999, 2011.
[11] L. Liu, R. Zhang, and K. Chua, “Wireless information transfer with opportunistic energy harvesting,” submitted to IEEE
Transactions on Wireless Communications, http://arxiv.org/abs/1204.2035, 2012.
[12] K. Huang and E. G. Larsson, “Simultaneous information-and-power transfer for broadband downlink systems,” submitted
to IEEE ICASSP 2013, http://arxiv.org/abs/1211.6868, 2012.
[13] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Relaying protocols for wireless energy harvesting and information
processing,” submitted to IEEE Transactions on Wireless Communications, http://arxiv.org/abs/1212.5406, 2012.
[14] K. Tutuncuoglu and A. Yener, “Sum-rate optimal power policies for energy harvesting transmitters in an interference
channel,” Journal of Communications and Networks, vol. 14, no. 2, pp. 151–161, Apr. 2012.
May 1, 2013
DRAFT
28
R−E tradeoff curves when Mt=3, Mr=4
200
Max. energy beamforming
Min. leakage beamforming
SLNR max. beamforming [22], [23]
SLER max. beamforming
180
160
Energy (Joule/sec)
140
120
100
80
rank(Q1)=1
60
40
No transmission at 1st Tx.
rank(Q1)=0
20
0
Fig. 7.
8.5
9
9.5
10
Rate (bits/s/Hz)
10.5
11
11.5
R-E tradeoff curves for MEB, the MLB, SLNR maximizing beamforming, and SLER maximizing beamforming when
Mt = 3 and Mr = 4.
R−E tradeoff curves for SLER max. beamforming when M=2
R−E tradeoff curves for SLER max. beamforming when M=2
Without SLER based scheduling, αij=0.7, i≠ j
180
Without SLER based scheduling, αij=1
180
With SLER based scheduling, αij=1
160
160
140
140
120
120
Energy (Joule/sec)
Energy (Joule/sec)
With SLER based scheduling, αij=0.7, i≠ j
100
80
100
80
rank(Q )=1
1
rank(Q1)=1
60
60
40
40
No transmission at 1st Tx.
rank(Q )=0
1
No transmission at 1st Tx.
rank(Q )=0
20
20
1
0
5.5
6
6.5
7
Rate (bits/s/Hz)
7.5
(a)
Fig. 8.
8
8.5
9
0
5.5
6
6.5
7
7.5
Rate (bits/s/Hz)
8
8.5
9
(b)
R-E tradeoff curves for SLER maximizing beamforming with/without SLER-based scheduling when (a) αij = 0.7
and (b) αij = 1 for i 6= j. Here, M = 2.
May 1, 2013
DRAFT
29
[15] ——, “Transmission policies for asymmetric interference channels with energy harvesting nodes,” in Proc. IEEE
International Workshop on Computational Advances in Multi-sensor Adaptive Processing, 2011, Dec. 2011, pp. 197–200.
[16] K. Huang and V. K. N. Lau, “Enabling wireless power transfer in cellular networks: architecture, modeling and deployment,”
submitted to IEEE Transactions on Signal Processing, http://arxiv.org/abs/1207.5640, 2012.
[17] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer: architecture design and rate-energy tradeoff,”
submitted to IEEE Transactions on Communications, http://arxiv.org/abs/1205.0618, 2012.
[18] C. Shen, W. Li, and T. Chang, “Simultaneous information and energy transfer: A two-user MISO interference channel
case,” in Proc. IEEE GLOBECOM, 2012, Dec. 2012.
[19] G. Scutari, D. P. Palomar, and S. Barbarossa, “The MIMO iterative waterfilling algorithm,” IEEE Trans. Signal Processing,
vol. 57, no. 5, pp. 1917–1935, May 2009.
[20] W. Yu, W. Rhee, S. Boyd, and J. M. Cioffi, “Iterative water-filling for gaussian vector multiple-access channels,” IEEE
Trans. Inform. Theory, vol. 50, no. 1, pp. 145–152, Jan. 2004.
[21] A. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath, “Capacity limits of MIMO channels,” IEEE J. Select. Areas
Commun., vol. 21, no. 5, pp. 684–702, June 2003.
[22] M. Sadek, A. Tarighat, and A. H. Sayed, “A leakage-based precoding scheme for downlink multi-user MIMO channels,”
IEEE Trans. Wireless Commun., vol. 6, no. 5, pp. 1711–1721, May 2007.
[23] J. Park, J. Chun, and H. Park, “Generalised singular value decomposition-based algorithm for multi-user MIMO linear
precoding and antenna selection,” IET Commun., vol. 4, no. 16, pp. 1899–1907, Nov. 2010.
[24] S. L. Loyka, “Channel capacity of MIMO architecture using the exponential correlation matrix,” IEEE Commun. Lett.,
vol. 5, no. 9, pp. 369–371, Sept. 2001.
[25] C. C. Paige and M. A. Saunders, “Towards a generalized singular value decomposition,” SIAM Journal on Numerical
Analysis, vol. 18, no. 3, pp. 398–405, June 1981.
[26] D. A. Harville, Matrix algebra from a statistician’s perspective.
Berlin: Springer, 2008.
[27] R. A. Horn, N. H. Rhee, and W. So, “Eigenvalue inequalities and equalities,” Linear Algebra and Its Applications, vol.
270, no. 1-3, pp. 29–44, Feb. 1998.
[28] V. R. Cadambe and S. A. Jafar, “Interference alignment and degrees of freedom of the K-user interference channel,” IEEE
Trans. Inform. Theory, vol. 54, no. 8, pp. 3425–3441, Aug. 2008.
[29] S. Boyd and L. Vandenberghe, Convex Optimization, 7th ed.
New York: Cambridge University Press, 2009.
[30] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed.
New Jersey: John & Sons, Inc., 2006.
[31] X. Zhao, P. B. Luh, and J. Wang, “Surrogate gradient algorithm for Lagrangian relaxation,” Journal of Optimization Theory
and Applications, vol. 100, no. 3, pp. 699–712, Mar. 1999.
[32] J. Park, J. Chun, and B. Jeong, “Efficient multi-user MIMO precoding based on GSVD and vector perturbation,” Signal
Processing, vol. 92, no. 2, pp. 611–615, Feb. 2012.
[33] R. H. Gohary, W. Mesbah, and T. N. Davidson, “Rate-optimal MIMO transmission with mean and covariance feedback at
low SNRs,” IEEE Trans. Veh. Technol., vol. 58, no. 7, pp. 3802–3807, Sept. 2009.
[34] S. A. Jafar and A. Goldsmith, “Transmitter optimization and optimality of beamforming for multiple antenna systems,”
IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1165–1175, July 2004.
[35] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. Wireless
Commun., vol. 9, no. 11, pp. 3590–3600, Nov. 2010.
May 1, 2013
DRAFT
30
[36] J. Hoydis, S. T. Brink, and M. Debbah, “Massive MIMO: How many antennas do we need?” in Proc. Annual Allerton
Conference on Communication, Control, and Computing, 2011, Sept. 2011, pp. 545–550.
[37] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO:
opportunities and challenges with very large arrays,” IEEE Signal Processing Mag., vol. 30, no. 1, pp. 40–60, Jan. 2013.
[38] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed.
Boston: McGraw-Hill,
2002.
May 1, 2013
DRAFT